Permittivity

Permittivity $ϵ$ is the material parameter that links the Electric flux density $\mathbf{D}$ to the Electric field $\mathbf{E}$:

$\mathbf{D}=ϵ\mathbf{E}.$

Units: farads per meter (F/m). The split

$ϵ={ϵ}_r{ϵ}_0$

separates the medium’s effect (${ϵ}_r$, relative permittivity, dimensionless, also called the dielectric constant) from the universal constant of free space, ${ϵ}_0=8.854\times 10^{-12}$ F/m.

For vacuum: ${ϵ}_r=1$. For air: ${ϵ}_r\approx 1.0006$, near enough to 1 that it’s usually treated as vacuum. For typical dielectrics: ${ϵ}_r$ ranges from 2 (PTFE) to ~10 (glass, mica) to ~80 (water) to thousands (ferroelectric ceramics).

Why a medium has permittivity bigger than 1

When an external field $\mathbf{E}$ is applied, the medium becomes polarized: bound charges (electron clouds shifting against nuclei) line up against the field, producing internal dipoles. These dipoles create their own field opposing the external one, partially cancelling it. The net effect: the same free-source flux density $\mathbf{D}$ produces a smaller total field $\mathbf{E}=\mathbf{D}/ϵ$ inside the material.

The microscopic relation:

$\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P},$

with linear polarization $\mathbf{P}={ϵ}_0{\chi }_e\mathbf{E}$. Comparing with $\mathbf{D}=ϵ\mathbf{E}$ gives

$ϵ={ϵ}_0(1+{\chi }_e),{ϵ}_r=1+{\chi }_e,$

where ${\chi }_e$ is the electric susceptibility of the medium.

Free-space permittivity in context

${ϵ}_0$ sits in Coulomb’s law as the natural unit-fixing constant. Together with ${\mu }_0$ (the Permeability of free space) it sets the speed of light:

$c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}\approx 3\times 10^8\text{ m/s}.$

This is one of the deepest unifications in physics — the static-electricity constant and the static-magnetism constant together fix the speed of electromagnetic waves.

Linear, isotropic, homogeneous

Electromagnetics assumes (for most material) that the medium is

• Linear: $\mathbf{P}$ is proportional to $\mathbf{E}$ (no saturation, no nonlinear optics).
• Isotropic: $\mathbf{P}$ is parallel to $\mathbf{E}$ — direction-independent. (In crystals it isn’t, and $ϵ$ is a tensor.)
• Homogeneous: $ϵ$ is constant in space.

Under these conditions, $ϵ$ is a single scalar, and $\mathbf{D}=ϵ\mathbf{E}$ holds everywhere. The notes use this assumption unless explicitly noted.

Dielectric strength

Even an ideal dielectric breaks down if the field becomes too large — see Dielectric breakdown. The dielectric strength $E_{\text{ds}}$ is the maximum field the material can sustain before ionizing and conducting. It is unrelated to ${ϵ}_r$ — air has small ${ϵ}_r$ but moderate $E_{\text{ds}}$; oil-impregnated paper has larger ${ϵ}_r$ and higher $E_{\text{ds}}$.

In wave propagation

In time-varying fields, $ϵ$ governs the wave speed inside the medium:

$u_p=\frac{c}{\sqrt{{ϵ}_r}}\text{(non-magnetic medium)}.$

This shows up directly in Transmission line analysis — the phase velocity on a TEM line filled with dielectric of relative permittivity ${ϵ}_r$ is $c/\sqrt{{ϵ}_r}$. It’s why a 1-meter coax cable behaves electrically like a longer line in free space: signals are slower inside.